The US Supreme Court has "the supreme power" because the Framers of the Constitution stated in Article III, Section 1, that "The judicial power of the United States, shall be vested in one Supreme Court, and in such inferior courts as the Congress may from time to time ordain and establish."
It was necessary to make one body the ultimate authority on federal and constitutional law and treaties to establish uniform interpretation of the laws so the federal court system would make consistent decisions. This intentionally differed from English common law in that the English system of that time allowed each court to set its own precedents, creating a situation where laws could be applied differently or unfairly, depending on where a person was tried.
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the supreme court uses the judicial review to check the powers of the legislative and executive branches
Article III deals with the supreme courts powers.
the ability to say no
One power would be judicial review, which the supreme court uses to not only declare laws unconstitutional, but also the actions of the President or Congress. Another power would be the ability to interpret the constitution.
Had Congress allowed six more justices on the Supreme Court, how might have this changed the balance of powers